The periodic pattern in Fig. REF a
is similar to the experimental observation of a periodic array of
flux-closure domains in PbTiO\(_{3}\)  reported by Tang et
al. [1]} While with \(s=7.1\mathrm {nm}\) ,
see Fig.Â REF b a herringbone domain pattern is stabilised.
Care is needed in interpreting these results: observe that scaling
of the cell size in these simulations (or equivalently, changing the
domain spacing) leads to different equilibrium states. This happens
because the imposed conditions have forced the polarization, at first,
far from the spontaneous states making the subsequent evolution of
domains highly sensitive to the kinetics, here represented crudely
through the polarization viscosity. Thus the path of domain evolution
during switching is to some extent an artefact of the model. Also,
from an energetic point of view, the nucleation of the new domains
is less costly in the simulation if the pattern adopts the minimum
number of periods that will fit into the simulation cell. This explains
the way the individual domains form new layers with the minimum possible
spatial frequency. This is an artefact of the cell size: in nature
there is no imposed cell and the new spacing must arise purely from
a balance between the gradient and bulk energies. Nevertheless, the
simulations in Fig. REF  indicate the opportunity to engineer
complex patterns such as the rank-2 laminates by starting with a simple
rank-1 pattern and applying suitable external conditions that force
the pattern to nucleate new domains. The length-scale of the initial
pattern then controls or guides the generation of the new pattern.
We have given only a few examples to illustrate the process, but expect
that similar methods could enable the generation of a range of patterns. 
